where \(\chi _{\underline{n}}\) , \(\underline{n}\in {\mathbb {N}}^s\) , are appropriate unit modulus eigenfunctions of the system \((X,\mu ,T)\) .
In Steps 2 and 3, using suitable manipulations of (REF ), the good equidistribution properties of the sequences \(a_1,\ldots , a_\ell \) , and the induction hypothesis, we deduce that the eigenfunctions \(\chi _{\underline{n}}\)  satisfy some algebraic relations, which allow us to conclude that they are products of sequences in \(s-1\)  variables.We crucially use in this step that the sequences \(a_1,\ldots , a_\ell \)  are good for equidistribution; if \(\ell \ge 3\)  and we only knew that they were good for irrational equidistribution (for example if they were polynomial sequences), then our argument would have been much more complicated (as is the case in [1]}). In view of this, (REF ) combined with Lemma REF  enables us to show that
\(\vert \!|\!| \tilde{f}_l|\!|\!\vert _{s+1}>0\) . Iterating this step \(s+1\)  times we deduce that \(\int \tilde{f}_l\, d\mu >0\) . Given this, it is an easy matter in Step 4 to contradict the assumptions of Proposition REF .
